Fitting parameters and therapies of ODE tumor models with senescence and immune system

This work is devoted to introduce and study two quasispecies nonlinear ODE systems that model the behavior of tumor cell populations organized in different states. In the first model, replicative, senescent, extended lifespan, immortal and tumor cells are considered, while the second also includes immune cells. We fit the parameters regulating the transmission between states in order to approximate the outcomes of the models to a real progressive tumor invasion. After that, we study the identifiability of the fitted parameters, by using two sensitivity analysis methods. Then, we show that an adequate reduced fitting process (only accounting to the most identifiable parameters) gives similar results but saving computational cost. Three different therapies are introduced in the models to shrink (progressively in time) the tumor, while the replicative and senescent cells invade. Each therapy is identified to a dimensionless parameter. Then, we make a fitting process of the therapies’ parameters, in various scenarios depending on the initial tumor according to the time when the therapies started. We conclude that, although the optimal combination of therapies depends on the size of initial tumor, the most efficient therapy is the reinforcement of the immune system. Finally, an identifiability analysis allows us to detect a limitation in the therapy outcomes. In fact, perturbing the optimal combination of therapies under an appropriate therapeutic vector produces virtually the same results. Supplementary Information The online version contains supplementary material available at 10.1007/s00285-023-02000-9.


Supplementary material A Initial and final values
In this Supplementary Material section, we provide the information about the initial and the final state of each population (as well as the starting time t ⇤ of therapies) for each model, with or without therapy.

A.1 Tumor Invasion
The value at the final time of each population for each model using the optimal parameter combinations (calculated for all parameters at Section 2 and only for the identifiable parameters at Subsection 4.2) are given in Table 9: y 2 (T ) y 2 (T ) y 2 (T ) y 3 (T ) y 3 (T ) y 3 (T ) y 4 (T ) y 4 (T ) y 4 (T ) y 5 (T ) y 5 (T ) y 5 (T ) y 6 (T ) y 6 (T ) y 6 (T ) 5 5 5 Param.5 5 5 Pop.5.2782

A.2 Therapies
The value at the initial time t ⇤ at which therapies were initiated and the value of each population at t ⇤ , for both models and for each tumor initial value studied in Section 5 are listed in Table 10.Also, the value at the final time of each population of each model and for each tumor initial value using the optimal combination of parameters are given in Table 11.Table 11: Final values of the populations for the optimal therapy parameters. ( Table 12: Final values of the populations for the optimal values of the new therapy parameters for the 5-populations model.

A.3.2 6-populations model
The results are similar to the 5-populations model.

B Sensitivity analysis
We study the derivatives of the populations with respect to the parameters.

B.1 Tumor invasion
Figures 10 and 11 show the derivatives with respect to the parameters {Q 1 , Q 3 , Q 4 , ↵, }, which lead us to the tumor invasion at their optimal values, for the 5 and 6-populations models, respectively (it is important to note not only the shape but also the scale of the graphics).
Looking at Figures 10 and 11, we can deduce that the derivatives with respect to Q 4 and Q 3 are rather small (showing the little identifiability of parameters Q 4 and Q 3 ) whereas derivatives with respect to ↵ and are dominant.
We can appreciate that all the derivatives have a similar shape, presenting the greatest variation at the beginning of the tumor invasion and showing that an increase of any of these parameters benefits the replicative population (and to a lesser extent the senescent Table 13: Final values of the populations for the optimal values of the new therapy parameters for the 6-populations model. population) whereas it is detrimental for the tumor population.It also shows that there is a transient stability before the tumor invasion where increasing the parameters increases the replicative population while the senescent population decreases but to a lesser extent than the other (also only seen in the Q 1 graphic).

B.2 Therapies
We show the derivatives with respect to therapy parameters for each model.

B.2.1 5-populations model
The derivatives of the populations with respect to each therapy parameter for this model at each tumor initial value can be seen in Figure 12.
Observe that the scale for both parameters increases with the tumor initial value and the derivatives with respect to parameter  1 have a greater scale than with respect to the parameter  2 in all cases.
• Increasing the parameter  1 benefits the replicative population while it is detrimental mainly for the tumor and slightly to the senescent one (which is even less the greater is the tumor initial value), increasing to its absolute maximum value quickly and decreasing gradually over time in all cases.• Increasing the parameter  2 causes a similar e↵ect on the populations as  1 except for the senescent cells, which initially feels beneficial (which remains in time longer the greater the tumor initial value is) and, at a certain time, it starts to decrease, reaching negative values but in lesser extent the greater is the tumor initial value.

B.2.2 6-populations model
The derivatives of the populations with respect to each therapy parameter at each tumor initial value can be seen in Figure 13.
We can deduce that the scale for all the parameters increases with respect to the initial tumor (except for  2 , whose scale decrease for initial tumor ⇡ 70%).
• Increasing the parameter  1 causes the following e↵ects: -For the replicative population, the derivative increases all the time except at the end, when the benefit becomes pronounced.The higher the initial tumor, the earlier the change in trend occurs, and it also occurs after the benefits start to wane and disappear.
-For the senescent population, it causes di↵erent behaviors over time (some of which can only be observed when the initial tumor is large enough).Initially, it causes a small decrease but will eventually disappear and then a slightly larger increase will start but will also go away eventually.
-For the tumor population, the behavior is similar to that of the senescent population, but always remaining below, highlighting the case of low initial tumor where its detrimental e↵ect is considerable, contrasting with the benefit for the replicative population, except for the last moments, when the immune population replaces the tumor.
-For the immune population, it causes a detrimental e↵ect, but for low initial tumor its abrupt change occurs only at the end and the bigger the initial tumor is the earlier this abrupt change occurs, which reaches its limit and begins to decrease (precipitously too) until it disappears.
• Increasing the parameter  2 has, in form, practically the same e↵ects as the previous parameter  1 .
• Increasing the parameter  3 has no e↵ect on the populations at the beginning, but in the end it has a precipitous benefit for the immune population and an almost analogous e↵ect for the replicative but this time detrimental.It is also detrimental, to a lesser extent, for both senescent and tumor populations.Moreover, the higher the initial tumor, the earlier this strong variation occurs, reaching its limits and beginning to decrease until it disappears, except for the tumor population that remains a bit and the replicative one, which not only disappears its harmful e↵ect but also begins to be slightly beneficial.
The increase of the initial tumor also shortens the di↵erence between the detrimental e↵ect of replicative and tumor populations.

C Eigenvectors and Eigenvalues
We provide the eigenvectors and eigenvalues for each identifiability method and for each model, with or without therapy.

C.1 Tumor Invasion
As a consequence, the biological parameters, ordered from least identifiable to most identifiable are:

IX
So that, the biological parameters, ordered from least identifiable to most identifiable are: • Eigenvalues Method Consequently, the biological parameters, ordered from least identifiable to most identifiable are:

C.2.1 5-populations model
Making the corresponding change of variables from i to  i , we give the combinations of original therapy parameters  i in Table 14.
The conclusions obtained from the eigenvalues and eigenvectors are discussed in Subsection 6.1 and are as follows: • For the initial tumor value ⇡ 10%: -P CA P CA P CA Method * y 1 population:

Figure 10 :
Figure 10: Dynamical behavior of the derivatives of the five cell types for each transmission parameter.The upper graphic is for Q 1 , the middle-left one for Q 3 , the middle-right one for Q 4 , the bottom-left one for ↵ and the bottom-right one for .Line codes, y 1 : blue crossed, y 2 : red circled, y 3 : yellow diamonded, y 4 : purple squared, y 5 : green 5 points stared.

| 1 |
Figure 11: Dynamical behavior of the derivatives of the six cell types for each transmission parameter.The upper graphic is for Q 1 , the middle-left one for Q 3 , the middle-right one for Q 4 , the bottom-left one for ↵ and the bottom-right one for .Line codes, y 1 : blue crossed, y 2 : red circled, y 3 : yellow diamonded, y 4 : purple squared, y 5 : green 5 points stared, y 6 : cyan 6 points stared.

Table 10 :
Initial time and values of the populations when therapies are applied.
for y 3 .